NHBC 10-year is an insurance policy against major issues. There is a shorter (2-year I think) warranty on things having been done properly / to an acceptable standard.
I’ve snagged 2 of my own houses, and a flat for a family member. Generally I’d follow a basic system of working round each room systematically (e.g. always go round to the left from the doorway). Do a general check on finish of paintwork and woodwork. Anything that’s been installed (e.g. tiling, flooring, plumbing, heating, electrical sockets, doors, windows, cupboards) check the quality of installation (is it finished, straight, tight, tidy etc) and the function (does it work, is it smooth etc as appropriate). Check also that what is/was specified is actually installed. Do similar outside, as well as having a look inside loft spaces etc. To do this properly takes a while, and some familiarity with what/how things should be done helps, but isn’t necessarily essential.
As above, it’s not necessary to be a pain, but presenting something structured will help – most builders are used to verbal / fag packet lists only. If you’re organised then builders will initially be unsure how to take it if you hand over a printed list, but actually quite like it as it gives everyone a clear picture. I used to give them an updated version every couple of weeks showing what was still outstanding, and adding any initial ideas that crop up. Being reasonable/pragmatic in terms of how things can be fixed helps as well – if you’re sensible then the developer is more likely to respect your opinion and go out of their way to make you happy.
In the case of the flat I looked at for a family member the ‘premium’ developer involved was initially totally dismissive of there being any faults at all (as they had already ‘snagged’ it themselves). However, a well-presented list made them review the situation, and would have provded a sound basis for escalation had they contested things.